Inventory Manager is not reducing the inventory count when customers add products to their basket. Why?

There are a few possible reasons but here is one possiblity that is a good place to start:

In order to check on the inventory levels and correctly redirect the customer, the Inventory Manager registers itself with Merchant as a "logging" module. Normally, you would not notice this except that the Inventory Manager will be listed on the "Logging Configuration" screen in your store. However, in some versions of Merchant the module is listed there with an editable checkbox so that it can be "un assigned" as a logging module. That will keep the module from correctly interacting with the store. Go to the "Logging Configuration" screen and, if there is a checkbox next to "Inventory Manager" check to make sure that it is checked (aka "on"). In later versions of Merchant (I have Production Release 3) the merchant cannot un-check that box.